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Jackson and smoothness theorems for Freud weights in $$L_ p (0<p\leq\infty)$$. (English) Zbl 0867.41010
The authors study Jackson, realization and converse theorems for Freud weights in $$L_p$$ spaces, $$0<p \leq\infty$$. The method adopted here even for the case $$1\leq p\leq \infty$$ for Jackson theorems for Freud weights avoids the use of the deep properties of orthogonal polynomials. Some properties of the modulus of smoothness in the $$L_p$$ space for $$0<p\leq \infty$$ have been given involving realization functionals. The technique of first approximating by a spline and then by a polynomial as followed here seems to be new in the context and is of some intrinsic interest. Various other interesting results, such as some Marchaud-type inequalities, are also given.

MSC:
 41A10 Approximation by polynomials 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis